Reciprocal & Inverse Trigonometric Functions (AQA A Level Maths: Pure): Exam Questions

(i) For $-2\pi \le \theta \le 2\pi$, sketch the curve $y=3+2\mathrm{cosec}\theta$, showing the coordinates of any maximum and minimum points.

(ii) Hence, find the number of solutions to the equation

$3+2\mathrm{cosec}\theta =\frac{1}{2}$

in the interval  $-2\pi \le \theta \le 2\pi$.

Write $\tan \theta \mathrm{cosec}\theta$ as a single trigonometric function.

Hence solve for $-\pi <\theta \le \pi$

$\tan \theta \mathrm{cosec}\theta =-\frac{2\sqrt{3}}{3}$

Solve the equation

$\sec \theta \tan \theta -\sec \theta =0$

for $0\le x\le 2\mathrm{\pi }$, giving your answers in exact form.

Show that

$\cot x\mathrm{cosec}x\sec x\equiv 1+{\cot }^{2}x$

Solve the equation

${\mathrm{cosec}}^{2}x-2\mathrm{cosec}x-8=0$

for $0°\le x\le 360°$, giving your answers to 1 decimal place where appropriate.

Show that the equation

$3-\sec \theta =\frac{2}{\sec \theta }$

may be written in the form

 $\left(\sec \theta -2\right)$$\left(\sec \theta -1\right)=0$

Hence solve

$3-\sec \theta =\frac{2}{\sec \theta }$

for $0\le \theta \le 2\pi$,

Show that the equation

${\tan }^{2}x=6\sec x-10$

may be written in the form

 ${\left(\sec x-3\right)}^2=0$

Hence solve for $0\le x\le 2\pi$

${\tan }^{2}x=6\sec x-10$

giving your answers correct to 3 significant figures.

Sketch the graph of $y=2\sec 2x$ for $-\pi \le x\le \pi$.

Label the asymptotes and any maximum and minimum points.

Use the sketch in part (a) to find the number of solutions to the equation

$2\sec 2x=4$

in the interval  $-\pi \le x\le \pi$.

The function $\mathrm{f}$ is defined by

$\mathrm{f}\left(x\right)=\mathrm{arc}\cos x$

for $-1\le x\le 1$, and the function $\mathrm{g}$ is defined by

$\mathrm{g}\left(x\right)=\mathrm{f}\left(3x\right)$

Sketch the graph of $y=\mathrm{f}\left(x\right)$  and state the range of  $\mathrm{f}$.

Sketch the graph of $y=\mathrm{g}\left(x\right)$ and state the domain of $\mathrm{g}$.

Find ${\mathrm{g}}^{-1}\left(x\right)$ and state its domain.

Show that the equation

$2{\cot }^{2}x=1-5\mathrm{cosec}x$

can be written in the form

$\left(2\mathrm{cosec}x-1\right)\left(\mathrm{cosec}x+3\right)=0$

Hence solve, for $0\le x\le 2\pi$, the equation

$2{\cot }^{2}x=1-5\mathrm{cosec}x$

giving your answers correct to 3 significant figures.

Solve, for $0\le \theta \le 2\pi$, the equation

$\frac{2}{\mathrm{cosec}\theta }-\mathrm{cosec}\theta =1$

Given that $x$ satisfies the equation $\arccos x=k$ , where   $0<k<\frac{\pi }{2}$.

(i) state the range of possible values of $x$,

(ii) find both $\sin k$ and $\tan k$ in terms of $x$.

Show that the equation

$\sec x\mathrm{cosec}x-5=\mathrm{cosec}2x$

may be written in the form

$\mathrm{cosec}2x=5$

Hence solve for $0\le x\le 2\pi$

$\sec x\mathrm{cosec}x-5=\mathrm{cosec}2x$

giving your answers correct to 3 significant figures.

Given that $x$ satisfies the equation  $\mathrm{arc}\sin$ $x=k$, where  $-\frac{\pi }{2}<k<0$  ,

(i) state the range of possible values of $x$,

(ii) find both $\cos k$ and $\tan k$ in terms of $x$.

The function $\mathrm{f}$ is defined by

$\mathrm{f}\left(x\right)=\arctan x$

where  $x\in \mathrm{ℝ}$ , and the function $\mathrm{g}$ is defined by

$\mathrm{g}\left(x\right)=\frac{2}{\mathrm{\pi }}\mathrm{f}\left(x\right)-1$ 

Sketch the graph of $y=\mathrm{f}\left(x\right)$ and state the range of $\mathrm{f}$.

Sketch the graph of $y=\mathrm{g}\left(x\right)$and state the range of $\mathrm{g}$.

Find ${\mathrm{g}}^{-1}\left(x\right)$ and state its domain.

(i) Sketch, for $-2\pi \le \theta \le 2\pi$, the curve $y=-5+\frac{1}{2}\sec \theta$.

Label clearly the coordinates of any maximum and minimum points.

(ii) Hence find the range of values of $k$ for which the equation   

$-5+\frac{1}{2}\sec \theta =k$

has no solutions.

Solve for $-\pi <\theta \le \pi$

$\frac{\sec \theta \cot \theta }{\mathrm{cosec}\theta \tan \theta }=-\sqrt{3}$

Given that $x$ satisfies the equation  $\arctan x=k$ where  $-\frac{\pi }{2}\le k\le 0$

(i) state the range of possible values of $x$,

(ii) find both $\sin k$ and $\cos k$ in terms of $x$.

Solve for $-\pi <x\mathit{\le }\pi$

$\left(\mathrm{cosec}x\mathit{-}\sec x\right)\left(\frac{1}{\sec x}+\frac{1}{\mathrm{cosec}x}\right)=\cot 2x+3$

Give your answers correct to 3 significant figures.

Solve

$\sec x\mathrm{cosec}x-75=5\mathrm{cosec}2x$

for $-\pi <x\mathit{\le }\pi$.  

Give your answers correct to 3 significant figures.

Solve for $0\le \theta \le 2\pi$

$6\sec \theta +\frac{2\sqrt{3}}{\sec \theta }=-3-4\sqrt{3}$

Leaving your answers exact.

Solve for $0\le x\le 2\pi$

$3{\cot }^{2}x-4\sqrt{3}=\left(6-2\sqrt{3}\right)\mathrm{cosec}x-3$

giving your answers in an exact form.

Sketch, for $-2\pi \le \theta \le \pi$, the curve $y=2+3\sec \left(\theta +\frac{\mathrm{\pi }}{2}\right)$ .

Label clearly the coordinates of any maximum and minimum points.

Find the maximum and minimum values of

$\frac{1}{2+3\sec \left(\theta +{\displaystyle \frac{\mathrm{\pi }}{2}}\right)}$

The function $\mathrm{f}$ is defined by

$\mathrm{f}\left(x\right)=\arcsin x$

for  $-1\le x\le 1$ , and the function $\mathrm{g}$ is defined by 

$\mathrm{g}\left(x\right)=\frac{4\mathrm{f}\left({\displaystyle \frac{x}{3}}\right)}{\mathrm{\pi }}+2$

Sketch the graph of $y=\mathrm{g}\left(x\right)$and state the domain and range of $\mathrm{g}$.

Find ${\mathrm{g}}^{-1}\left(x\right)$ and state its domain.

Over the same domain as $\mathrm{g}$, the function $\mathrm{h}$ is defined by

$\mathrm{h}\left(x\right)=p\arccos \left(\mathrm{q}x\right)$

Given that $\mathrm{h}\left(x\right)=-\mathrm{g}\left(x\right)$ for all $x$ in the domain, find the values of $p$ and $q$.

Show that for $0\le x\le 1$

$\mathrm{arc}\sin$ $x=\mathrm{arc}\cos$$\sqrt{1-x^2}$

Explain why the relationship in part (a) is not true for $-1\le x<0$.